Theorem. Let G be a simple, simply connected Lie group. Suppose that p is a prime that does not divide any of the coefficients of the highest root (expressed as a linear combination of the simple roots). Then $H^*G$ has no p-torsion.

Here $H^*G$ refers to the integral cohomology of $G$ as a space.

(Interestingly, the converse of this statement does not hold. For example $Sp(n)$ does not have $2$-torsion, though the coefficients of the highest weight are (1,2). This counterexample is given by Borel).

You can check out his proof here if you're interested, but basically it seems like what he does is this: he computes the cohomology of each of the simple, simply connected Lie groups with coefficients that have the relevant primes inverted and proves the result by observation.

Since the statement of the theorem itself has little to do with the actual classification of simple Lie groups, and morally should only rely on the fact that we can recover a simple, simply-connected Lie algebra from its root system, it seems natural to ask:

Question: In the intervening 50+ years since Borel proved this theorem, do we know of a direct proof of the theorem that does not use the classification of simple Lie groups?

I really have no idea how to go about doing this... could one use Lie algebra cohomology? I can only seem to find a link between Lie algebra cohomology and deRham cohomology of Lie groups, which is of no help when asking questions of torsion. Maybe there's an integral version?

You may want to look at math.ku.dk/~jg/papers/classificationpodd.pdf where we summarize what is known and unknown about this and related questions, and give references to the literature. See in particular Rem 10.9 and the onwards references from there.
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Jesper GrodalJun 14 '12 at 12:54

@Jesper: This was actually really helpful, if you feel up to taking the relevant bits and writing them as an answer I would accept it (and I'm sure others would appreciate it.) If not, I still found the paper quite helpful- and interesting!
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Dylan WilsonJun 15 '12 at 1:34

1 Answer
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I can't answer this completely, but I can point out some important follow-ups in the literature by Borel and others which need to be be taken into account. Borel's Tohoku paper is reprinted in the second volume of his collected papers (Springer, 1983). At the end of this volume is a full two page commentary by Borel on this paper, which reiterates his disappointment that he was unable to prove his result on torsion without going through the classification. Here he assembles more detailed results on "torsion primes" which were afterwards proved by Demazure (1973 Inventiones paper) and especially by Steinberg (1975 Advances paper). Eventually all of this work yields a clearer picture with less reliance on the classification.

Further closely related work then appeared in an AMS Memoir which Steinberg reviewed at length in Mathematical Reviews:

ADDED: It's a natural problem in Lie theory to find classification-free proofs of results which are first observed case-by-case. But I'm not aware of any substantial progress beyond Borel's careful commentary (in the 1983 volume), where he notes the equivalence of five statements about cohomology and the equivalence of three statements about orimes related to root systems and Weyl groups (with reference to Demazure and Steinberg). As he observes, the first five imply the last three, but only case-by-case study shows the converse as well.

Along the way Borel also points out an improvement over his original Nagoya formulation: while the "bad" primes (possibly 2, 3, 5) are those dividing a coefficient of the highest root, the "torsion" primes are the more limited ones dividing a coefficient of the coroot of this highest root. For instance in type $G_2$, with respective short and long simple roots $\alpha, \beta$, the highest root is $3\alpha + 2\beta$ whereas its coroot is $\alpha^\vee + 2\beta^\vee$.

There is a long history of study of the topology of a semisimple Lie group (equivalent to the topology of a maximal compact subgroup), in which a reduction is made to study of the root system and its Weyl group: for example, the determination of Betti numbers in terms of exponents or degrees for the Weyl group (Chevalley, ICM 1950). But this kind of transition is rather subtle. In the study of torsion primes, a key role is played by subgroups of maximal rank in a compact Lie group, these being correlated with certain subdiagrams of the extended Dynkin diagram. Much of the technology recurs in the study of p-compact groups, as Jesper observes. But not everything is well understood conceptually.

Thank you; I managed to find the articles you mentioned and it seems like what Steinberg can do is connect up a lot of different definitions for what a "torsion prime" could be, with the exception (of course) of giving a new proof that all of these notions agree with torsion in cohomology. It's really intriguing that this result really seems to rely on the classification... do you know if there is even a heuristic or general idea of why this theorem should be true, independent of the classification? Are there directions of proof that we know don't work? This is all really neat- Thank you!
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Dylan WilsonJun 15 '12 at 1:38

This answer is wonderful, thank you! The addition spelled things out a little more clearly, and I think I roughly understand what the current state of affairs is. Cheers,
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Dylan WilsonJun 15 '12 at 21:01